The theorem which we prove is more general than what we have described, since it works with the primary decomposition of the minimal polynomial, whether or not the primes which enter are all of first degree.

In the primary decomposition theorem, it is not necessary that the vector space V be finite dimensional, nor is it necessary for parts ( A ) and ( B ) that P be the minimal polynomial for T.

This theorem is similar to the theorem of Kakutani that there exists a circumscribing cube around any closed, bounded convex set in Af.

According to Cauchy's functional equation theorem, the logarithm is the only continuous transformation that transforms real multiplication to addition.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

The debate is interesting enough, however, that it is considered of note when a theorem in ZFC ( ZF plus AC ) is logically equivalent ( with just the ZF axioms ) to the axiom of choice, and mathematicians look for results that require the axiom of choice to be false, though this type of deduction is less common than the type which requires the axiom of choice to be true.

In Martin-Löf type theory and higher-order Heyting arithmetic, the appropriate statement of the axiom of choice is ( depending on approach ) included as an axiom or provable as a theorem.

Assuming ZF is consistent, Kurt Gödel showed that the negation of the axiom of choice is not a theorem of ZF by constructing an inner model ( the constructible universe ) which satisfies ZFC and thus showing that ZFC is consistent.

Assuming ZF is consistent, Paul Cohen employed the technique of forcing, developed for this purpose, to show that the axiom of choice itself is not a theorem of ZF by constructing a much more complex model which satisfies ZF ¬ C ( ZF with the negation of AC added as axiom ) and thus showing that ZF ¬ C is consistent.

It is possible, however, that there is a shorter proof of a theorem from ZFC than from ZF.

** Tarski's theorem: For every infinite set A, there is a bijective map between the sets A and A × A.

** König's theorem: Colloquially, the sum of a sequence of cardinals is strictly less than the product of a sequence of larger cardinals.

** Tychonoff's theorem stating that every product of compact topological spaces is compact.

** If S is a set of sentences of first-order logic and B is a consistent subset of S, then B is included in a set that is maximal among consistent subsets of S. The special case where S is the set of all first-order sentences in a given signature is weaker, equivalent to the Boolean prime ideal theorem ; see the section " Weaker forms " below.

Antiderivatives are important because they can be used to compute definite integrals, using the fundamental theorem of calculus: if F is an antiderivative of the integrable function f, then:

Using the above theorem it is easy to see that the original Borsuk Ulam statement is correct since if we take a map f: S < sup > n </ sup > → ℝ < sup > n </ sup > that does not equalize on any antipodes then we can construct a map g: S < sup > n </ sup > → S < sup > n-1 </ sup > by the formula

* The Lusternik – Schnirelmann theorem: If the sphere S < sup > n </ sup > is covered by n + 1 open sets, then one of these sets contains a pair ( x, − x ) of antipodal points.

The Bolzano – Weierstrass theorem gives an equivalent condition for sequential compactness when considering subsets of Euclidean space: a set then is compact if and only if it is closed and bounded.

If the ideals A and B of R are coprime, then AB = A ∩ B ; furthermore, if C is a third ideal such that A contains BC, then A contains C. The Chinese remainder theorem is an important statement about coprime ideals.

This is because if a group has sectional 2-rank at least 5 then MacWilliams showed that its Sylow 2-subgroups are connected, and the balance theorem implies that any simple group with connected Sylow 2-subgroups is either of component type or characteristic 2 type.

If X < sub > k </ sub > and Y < sub > k </ sub > are the DFTs of x < sub > n </ sub > and y < sub > n </ sub > respectively then the Plancherel theorem states:

Image: pons_asinorum. png | The bridge of asses theorem states that if A = B then C = D.

Image: Thales ' Theorem Simple. svg | Thales ' theorem: if AC is a diameter, then the angle at B is a right angle.

Thales ' theorem, named after Thales of Miletus states that if A, B, and C are points on a circle where the line AC is a diameter of the circle, then the angle ABC is a right angle.

In the Cartesian approach, the axioms are the axioms of algebra, and the equation expressing the Pythagorean theorem is then a definition of one of the terms in Euclid's axioms, which are now considered theorems.

If the conjecture were true, it would be a generalization of Fermat's last theorem, which could be seen as the special case n = 2: if, then.

If time, space, and energy are secondary features derived from a substrate below the Planck scale, then Einstein's hypothetical algebraic system might resolve the EPR paradox ( although Bell's theorem would still be valid ).

The argument above began by giving an unavoidable set of five configurations ( a single vertex with degree 1, a single vertex with degree 2, ..., a single vertex with degree 5 ) and then proceeded to show that the first 4 are reducible ; to exhibit an unavoidable set of configurations where every configuration in the set is reducible would prove the theorem.

To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn – Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable.

Rader's algorithm, exploiting the existence of a generator for the multiplicative group modulo prime, expresses a DFT of prime size as a cyclic convolution of ( composite ) size, which can then be computed by a pair of ordinary FFTs via the convolution theorem ( although Winograd uses other convolution methods ).

To formally state, and then prove, the completeness theorem, it is necessary to also define a deductive system.

The completeness theorem says that if a formula is logically valid then there is a finite deduction ( a formal proof ) of the formula.

It is deduced from the model existence theorem as follows: if there is no formal proof of a formula then adding its negation to the axioms gives a consisten theory, which has thus a model, so that the formula is not a semantic consequence of the initial theory.

The compactness theorem says that if a formula φ is a logical consequence of a ( possibly infinite ) set of formulas Γ then it is a logical consequence of a finite subset of Γ.

This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deduction system then implies φ is a logical consequence of this finite set.

Gödel's original proof of the theorem proceeded by reducing the problem to a special case for formulas in a certain syntactic form, and then handling this form with an ad hoc argument.

If one integrates this picture, which corresponds to applying the fundamental theorem of calculus, one obtains Cavalieri's quadrature formula, the integral – see proof of Cavalieri's quadrature formula for details.

The intermediate value theorem states the following: If f is a real-valued continuous function on the interval b, and u is a number between f ( a ) and f ( b ), then there is a c ∈ b such that f ( c ) = u.

If F is continuous then under the null hypothesis converges to the Kolmogorov distribution, which does not depend on F. This result may also be known as the Kolmogorov theorem ; see Kolmogorov's theorem for disambiguation.

# If A is a Lebesgue measurable set, then it is " approximately open " and " approximately closed " in the sense of Lebesgue measure ( see the regularity theorem for Lebesgue measure ).

If a definite statement is believed plausible by some mathematicians but has been neither proved nor disproved, it is called a conjecture, as opposed to an ultimate goal: a theorem that has been proved.

If, however, the set of allowed candidates is expanded to the complex numbers, every non-constant polynomial has at least one root ; this is the fundamental theorem of algebra.

If the polygon can be drawn on an equally spaced grid such that all its vertices are grid points, Pick's theorem gives a simple formula for the polygon's area based on the numbers of interior and boundary grid points.

The key result is the structure theorem: If R is a principal ideal domain, and M is a finitely

An important example is the ring Z / nZ of integers modulo n. If n is written as a product of prime powers ( see fundamental theorem of arithmetic ):

If both a 0 % rate and 100 % rate of taxation generate no revenue, it follows from the extreme value theorem that there must exist at least one rate in between where tax revenue would be a maximum.

The theorem " If n is an even natural number then n / 2 is a natural number " is a typical example in which the hypothesis is that " n is an even natural number " and the conclusion is that " n / 2 is also a natural number ".

If the force between any two particles of the system results from a potential energy V ( r ) = αr < sup > n </ sup > that is proportional to some power n of the inter-particle distance r, the virial theorem takes the simple form

If we choose the volume to be a ball of radius a around the source point, then Gauss ' divergence theorem implies that

* If the highest frequency B in the original signal is known, the theorem gives the lower bound on the sampling frequency for which perfect reconstruction can be assured.

* If instead the sampling frequency is known, the theorem gives us an upper bound for frequency components, B < f < sub > s </ sub >/ 2, of the signal to allow for perfect reconstruction.

If C is a complete category, then, by the above existence theorem for limits, a functor G: C → D is continuous if and only if it preserves ( small ) products and equalizers.

If the distribution of matter in each body is spherically symmetric, then the objects can be treated as point masses without approximation, as shown in the shell theorem.

Fermat's little theorem states that if p is prime and a is coprime to p, then a < sup > p − 1 </ sup > − 1 is divisible by p. If a composite integer x is coprime to an integer a > 1 and x divides a < sup > x − 1 </ sup > − 1, then x is called a Fermat pseudoprime to base a.